Leverage Is Not Reach: A Control-Window Law for Single-Neuron Steering in Language Models

The Control-Window Framework

This paper introduces a predictive, dose-aware framework for single-neuron steering in large language models (LLMs). Rather than treating neuron interventions as binary or fixed-magnitude events, the author models them as a pharmacological dose-response relationship. The core insight is that the effect of a single neuron's write direction on the model's output can be reduced to a single control coordinate: the alignment between the residual stream and the neuron's write vector. By normalizing this dose against a coherence budget—defined as the ratio of the residual norm to the write norm—the author identifies a universal saturation curve that dictates how the model's internal state rotates toward the intervention.

The Control-Window Law

The study establishes that coherent control is possible if and only if the behavior-trigger threshold lies below the collapse ceiling. The collapse ceiling represents the point at which the model's output degenerates into repetitive or nonsensical text. Crucially, this ceiling can be predicted from the model's weights and a single generic forward pass, allowing researchers to forecast whether a specific neuron is controllable before performing an intervention. The trigger threshold, conversely, is determined by the specific behavior being targeted (e.g., refusal, language routing, or arithmetic operations) and the rollout horizon.

Why Leverage Is Not Reach

A significant contribution of this work is the distinction between a neuron's "leverage" and its "reach." Standard gradient-based attribution methods often fail to identify effective controllers because true behavioral gates frequently write off the readout axis, resulting in near-zero first-order gradients. The author demonstrates that a forward-only contrastive screen, calibrated by the control-window law, successfully identifies these controllers where traditional attribution fails. This provides a more efficient and accurate recipe for finding behavioral gates without requiring expensive backward passes.